A question about random signal process

[qslazio]

I've read in a document as follow:

if v(t)=S(t)+n(t) (eq.1) , where S(t)=A•cos(wc•t)

S(t) is the signal, n(t) is a gaussian noise process.

The document said
v(t) can be expressed by

v(t)=A•cos(wc•t) + x(t)•cos(wc•t) -y(t)•sin(wc•t) (eq.2)
where x(t) and y(t) are real, low pass, Gaussian processes with zero mean and equal variance.

However, it never gives the explanation.
Can anyone explain to me how can we convert eq.1 to eq.2?
Thanks a lot!!


BTW: the document also call x(t) as inpahse noise or amplitude noise and
y(t) as quadrature or "phase noise"

 

[claudiocamera]

Ciao!
Well, probably it is not just like what you wrote.

You are probably reading a document about performance of comunicattions systems in presence of noise, so you have the quoted equality in a AM receiver with a signal contamined with a gaussian noise in its input.

In this context, you can have those equality because the noise in the receiver can be represented by its band pass equivalent. In order to understand the context you should first study band pass equivalent noise which basically states that a band pass noise can be expressed by n(t) =nc(t)•cos(wc•t) +ns(t)•sin(wc•t) where nc(t) and ns(t) are low frequency noise signal. After peforming the operations in the AM receiver and simplifying the expression you will get the answer you are looking for.

Good luck!